"Ideally, a welfare program would be considered successful if the number of clients went down every year. Is this how current programs behave? Or do they consider themselves successful the more people use them? The larger they become?"

Might better to rate them by how many new "clients" they got, then subtract from this total the number that remain on the rolls for a long time- or returned to it. You could have say 6 months as base line beyond which you start subtracting. You of course would also have different stats- indicating how much quality of life has been improved by the govt agency. Also, after the agency has "finished", it retains it's first function, but moves on the improving the lower case type people in general- people newly arrived in country, disabled, undereducated, etc.

The problem of course is this requires good management- and govt and bureaucracies has very strong tendency to fall into bad management. Plus as general rule the bigger the organization the better the management you need. Or said differently if give control to smaller management entities, then generally it will be managed better. Now, enter competition, if you bunch separate management entities who have independent control- not strictly controlled from "higher command" then you can measure and find better ways of doing something- they will find different approaches. Next, have transparency so those involved in supporting it can get reliable information.

But, all the above is still Top Down type management. Or socialism. And what you probably want is more Bottom Up type management (which also could be described as communism- though that word has been so corrupted by so many clueless people with their rigid dogma that it just adds confusion). Such as, networking, entreprenaur style, small groups, grassroot type stuff, clubs, religious groups, non profit, etc.